Question

The odds in favour of India winning any cricket match is $2:3$. What is the probability that if India plays $5$ matches, it wins exactly $3$ of them?

• A. ${}^5{C}_3{\left(\frac{2}{5}\right)}^2{\left(\frac{3}{5}\right)}^3$
• B. ${}^5{C}_3{\left(\frac{2}{3}\right)}^2{\left(\frac{1}{3}\right)}^3$
• C. ${}^5{C}_3{\left(\frac{2}{5}\right)}^3{\left(\frac{3}{5}\right)}^2$
• D. ${}^5{C}_3{\left(\frac{2}{3}\right)}^2{\left(\frac{1}{3}\right)}^2$

Answer 1

Answer: (C)

${}^5{C}_3{\left(\frac{2}{5}\right)}^3{\left(\frac{3}{5}\right)}^2$

Since India wins any $3$ out of the $5$ matches it plays, the matches it wins can be chosen by ${}^5{C}_3$ ways.

Now, since the odds in favour of India winning any cricket match is $2:3$, the probability of Indian winning the match becomes $\frac{2}{5}$ and India losing the match becomes $\frac{3}{5}$.

$\Rightarrow$ The final probability becomes ${}^5{C}_3{\left(\frac{2}{5}\right)}^3{\left(\frac{3}{5}\right)}^2$